D₅₃×#2081 #27413025

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2081

Splittings

Fitting	C₅₃×#2081 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2081

Subgroups

	subgroup	G/H
center	C₂	#397×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2081	C₂
socle	C₂×C₅₃	C₂×#397
prime radical	C₂²×D₄×D₅₃	C₂²
hypercenter	#2081	D₅₃
hypocenter	C₅₃	C₂×#2081
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2081

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(2,7)(3,12)(4,5)(8,15)(9,10)(11,16);(1,2,5,7,6,10,4,9)(3,11,13,16,12,15,14,8);(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)

Conjugacy classes & character table

644 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 198501);

D53×#2081 #27413025